Measurements

Computes the outcome probabilities for a quantum measurement. The conditional probabilities are determined by the Born rule, $P(i|j) = \text{Tr}[\Pi_i \rho_j]$, where $\Pi_j$ is a POVM element and $\rho_j$ is a density matrix.

Measurement of a single Ket or State:

measure(Π :: POVM, ρ :: State) :: Probabilities
measure(Π :: POVM, ψ :: Ket) :: Probabilities
measure(Π :: PVM, ρ :: State) :: Probabilities
measure(Π :: PVM, ψ :: Ket) :: Probabilities

Measurement of an ensemble of Ket or State types:

measure(Π :: POVM, ρ_states :: Vector{<:State}) :: Conditionals
measure(Π :: POVM, ψ_kets :: Vector{<:Ket}) :: Conditionals
measure(Π :: PVM, ρ_states :: Vector{<:State}) :: Conditionals
measure(Π :: PVM, ψ_kets :: Vector{<:Ket}) :: Conditionals

Measurement{T} <: AbstractVector{T}

The abstract type representing a quantum measurement. A measurement consists of a complete set of possible outcomes.

is_povm( Π :: Vector; atol=ATOL :: Float64 ) :: Bool

Returns true if Π satisfies the following constraints

• The POVM is complete: sum(Π) == I
• Each POVM element is hermitian
• Each POVM element positive semi-definite

is_povm_element(M :: AbstractMatrix; atol=ATOL :: Float64) :: Bool

Returns true if matrix M satisfies the following constraints:

• M is Hermitian
• M is positive semi-definite

POVM( Π :: Vector{POVMel{T}} ) <: Measurement{T}
POVM( Π :: Vector{Matrix{T}} ) <: Measurement{T}

Positve-operator valued measures (POVMs) represent a general quantum measurement. Each POVM-element is a hermitian, positive-semidefinite matrix. The sum of all POVM-elements yields the identity matrix. The constructor, POVM(Π) throws a DomainError if the provided array of matrices, Π is not a valid POVM.

POVMel( M :: AbstractMatrix{T<:Number}; atol=ATOL :: Float64) <: Operator{T}

A POVM element. A DomainError is thrown if matrix M is not hermitian and positive semi-definite within absolute tolerance atol.

mirror_symmetric_qubit_3povm( θ :: Real ) :: POVM{Float64}

Constructs a POVM aligned with the three mirror symmetric qubit states. The first measurement is aligned with the $|0\rangle$ state and the remaining two are symmetric across the z-axis.

A DomainError is thrown if argument θ ∉ [π/4,π/2].

asymmetric_qubit_3povm( θ1::Real, θ2::Real ) :: POVM

Constructs a general non-orthogonal 3-element POVM.

Inputs:

• θ1 ∈ (0,π/2] or [-π/2,0), the angle element 2 makes with $|0\rangle$ state.
• θ2 ∈ [-π/2,0) or (0,π/2], the angle element 3 makes with $|0\rangle$ state.

A DomainError is thrown if θ1 and θ2 are not in the valid ranges. Note that one angle must be positive and the other negative.

trine_qubit_povm() :: POVM{Float64}

The POVM with elements parallel to the trine qubit states (see trine_qubit_kets).

sic_qubit_povm() :: POVM{Complex{Float64}}

The POVM with elements parallel to the symmetric informationally complete (SIC) qubit states (see sic_qubit_kets).

planar_symmetric_qubit_povm( n :: Int64 ) :: POVM{Float64}

Constructs an n-element POVM{Float64} from the planar_symmetric_qubit_states. Each state is multipled by a factor of 2/n to satisfy the completeness relation.

A DomainError is thrown if n ≥ 2 is not satisfied.

sqrt_povm(priors :: Probabilities, states :: Vector{<:State}) :: POVM

Returns the "pretty good" square-root povm for the given density operator states and priors.

naimark_dilation( Π::POVM )

Returns the dilated projectors which are equivalent to the provided POVM. During measurement, the measured state must be tensored with the ancilla.

PVM( Π :: Vector{Vector{T}}; atol=ATOL :: Float64 ) <: Measurement{T}

The concret type for a projector-valued measure. The projectors are represented as a set of orthonormal basis vectors

A DomainError is thrown if Π does not contain an orthonormal basis.